Competing Ionization and Dissociation: Extension of the energy-dependent frame transformation to the gerade symmetry of H$_2$

TL;DR

This work extends the energy-dependent frame transformation ($EDFT$) to the gerade symmetry of H2, enabling a unified description of competing ionization and dissociation channels within multichannel quantum defect theory (MQDT). It combines $EDFT$ with an enhanced Jungen-Ross (JR) formalism to construct a full, energy-dependent lab-frame scattering matrix that includes both ionization and dissociation pathways, validated against an exactly solvable 2D model. The approach maps body-frame information $K_{jj'}({\cal E},R)$ to the lab-frame $S^{\rm phys}(E)$, capturing resonances and dissociation dynamics crucial for dissociative recombination (DR). The results show strong agreement with nearly exact 2D $R$-matrix solutions and demonstrate a path toward first-principles DR descriptions in diatomic ions, with potential extensions to ab initio input and more complex systems.

Abstract

This article solves two major tasks that frequently arise in the theory of electron collisions with a target molecular cation. First, it extends the energy-dependent frame transformation treatment(EDFT), which is needed to map fixed-nuclei electron-molecule scattering matrices into an energy-dependent laboratory frame scattering matrix with vibrational channel indices. The EDFT mapping can now be carried out even when the target molecule possesses multiple low energy potential curves, significantly transcending previous applications. Secondly, it implements a method to extract the rest of the full lab-frame scattering matrix, i.e. the columns and rows describing input and/or output dissociation channels. The treatment is benchmarked in this article against the essentially exact solution of a refined two-dimensional model of the singlet gerade $Σ$ symmetry of H$_2$. Our tests demonstrate that the theory accurately maps fixed-nuclei scattering information, of the type provided by existing electron-molecule computer codes, into a laboratory-frame scattering matrix that includes both ionization and dissociation. This treatment can provide a general framework applicable to a broad class of electron collision processes involving diatomic target ions, suitable for an accurate description of challenging processes such as dissociative recombination.

Figures (11)

• Figure 1: Comparison between neutral H$_2$ potential curves obtained using our updated model (blue dashed lines) versus our EPJD 2022 EPJD_H2_Gerade_2022 model (grey lines), both made to reproduce the data of Wolniewicz and Dressler Wolniewicz_Dressler_JCP_1985$EF$, $GK$ and $H\bar{H}$ curves (red points). The image also contains the Wolniewicz and Dressler $O$ and $P$ curves (also red points) as well as the H$_2^+$ ion potentials for reference (black lines).
• Figure 2: Example of body-frame energy curves ${\cal E}_{\epsilon,k}(R) = \epsilon+V_{k}^+(R)$ parallel to the ionic potentials. The blue dashed line is an example of an open channel with $\epsilon=0.01$ a.u. and $k=0$. The green dashed line represents a closed channel with $\epsilon=-0.03$ a.u., $k=0$ and the red dashed line represents a closed channel with $\epsilon=-0.05$ a.u., $k=1$.
• Figure 3: The energy dependence of the quantum defect matrix $\eta_{jj'}({\cal E},R)$ at two $R$ values. The full curves are evaluated at $R=2$ bohr and the dashed curves at $R=3$ bohr. In this graph, energy equal to zero has the same meaning as in Fig. \ref{['fig-BOcurves']}.
• Figure 4: Vibrational energy levels in a.u. of H$_2^+$ versus level number $v$ for two different boundary conditions and both choices of the ionic potential curve $V^+_k$. The blue crosses represent the boundary condition ${\chi_{i}^{(x)}}{'}(R_0)=0$ and the red circles represent ${\chi_{i}^{(0)}}(R_0)=0$. For this figure, $R_0=9$ a.u. The green squares show the differences (multiplied by 50) between them. The development assumes that the vibrational eigenfunctions for the first $\sim$14 $k=0$ states are phase-standardized to be identical for the $(0)$ and $(x)$ solutions obeying different boundary conditions, i.e. $\chi_i^{(0)}(R) \approx \chi_i^{(x)}(R)$ for $v_i \lesssim 14$ and $R<R_0$.
• Figure 5: Comparison of partial dissociative recombination cross sections. The input channel is $i=\{v=0,j=0\}$ and the output channel is the $EF$ curve. The incoming electron energies span from zero to $(E_{1,0}-0.031\text{ eV})$.